Toffoli gate supplemented with Hadamard gate can simulate any multi-qubit quantum computing [1]}. Toffoli gate is also served as an essential part in quantum factoring algorithm [2]}, quantum search algorithm [3]}, quantum half-adder [4]}, quantum error correction [5]}, and quantum fault tolerance [6]}, etc. Much attention has been paid to the realization of Toffoli gate [7]}, [8]}, [9]}.
It has been confirmed theoretically that the optimal cost of a Toffoli gate is six CNOT gates [10]} or five two-qubit entangling gates [11]}. Such synthesis might be helpful to design complex quantum gate, but it makes the gate further susceptible to the environmental noise and increases the time scale of the system.
Without using the standard decomposition-based approach, early in 2006, Fiurášek [12]} first showed a three-photon polarization Toffoli gate with a success probability of 0.75% (approximately 1/133) using linear optics. Using higher-dimensional Hilbert spaces, Ralph et al. [13]} improved the success probability of a linear optical Toffoli gate to 1/72 in 2007, and this interesting probabilistic scheme was later experimentally demonstrated in 2009 [14]}. Recently, Liu et al. [15]} further enhanced the success probability of the Toffoli gate to 1/64. Additionally, in 2022, Li et al. [16]} experimentally demonstrated a path-based three-photon Toffoli gate with a success probability of 1/72. Many hybrid multiple degrees of freedom (DOFs) probabilistic and deterministic Toffoli gates were also reported in recent years [17]}, [18]}, [19]}, [20]}, [21]}.
